Partial Differential Phase Shift Keying - Theory and Motivation

Recently, many evidences demonstrate that partial Differential Phase Shift Keying (i.e., when the delay inside the Delay Interferometer is shorter than the symbol period) can partially compensate the signal deformation caused by spectrally narrowing the optical channel (by interleavers, add-drop elements, WDM filters, etc.). In this paper the source of this effect is investigated with numerical simulations and, to the best of our knowledge for the first time, analytically. We found that our analytical analysis matched the simulation results with high accuracy. Furthermore, a phenomenological relation, which relates the optimum Free Spectral Range to the channel bandwidth, was derived.

, that the impairments due to optical spectrum narrowing can be partially compensated by using a DI having a differential delay smaller than the symbol time slot, i.e., , resulting in a Free Spectral Range (FSR) of the DI to be larger than the symbol rate, i.e., FSR>B. Moreover, it was shown [13] that the FSR increment beyond the bit-rate generally improves the system's tolerance to chromatic dispersion.
Thus, to date, a DI with FSR>B seems to be the best demodulator for spectrally narrow optical lines. The problem is that the source of this effect is till not completely understood. This is quite puzzling because the agreement of the experiment with simulations demonstrates that all the sources of this effect were taken under consideration. The challenge in identifying the main cause is the complexity of the system. The system includes many components: linear and non linear, with and without noises, optical and electrical, and the relations between them are not trivial.
Aside from the fact that this problem has practical significance, it also appears as a The model. One way to proceed is to realize that this effect is independent of noise.
The BER is of course noise dependent but the characteristic optimal FSR vs. filter's BW is not. This realization reduces the problem to finding the FSR, which maximizes the eye-opening as a function of the BW.
For a wide filter any increment of the DI's FSR narrows the eye-pattern and clearly deteriorates the system's performance. In Fig.1 we illustrate this point for BW=3B. By examining carefully the contribution of the different structures of a random sequence to narrowing the eye we find that the eye-opening is basically determined by two structures.
The dynamics of the minima of the eye-pattern. The minima of the EO is determined by a section of the sequence, which oscillates between two successive symbols, i.e., ( ) ( ) After passing through the filter the signal resembles a harmonic function (see Figs.3 and 4).
For simplicity we choose a Gaussian filter: Where f ∆ is the FWHM of the filter and for convenience matters we use . We chose Gaussian filters since our analysis and the result of many simulations show that the differences between the filters have a minor impact on the final curve. Commercial filters usually have a super Gaussian shape, but it will be shown that despite the simplicity of the Gaussian filters, the final curve agree very well with experimental and simulation results.
Due to the filter we can take, with great accuracy, only the two first harmonics, which the sequence is consisted of: This optical signal after passing through the DLI and then being detected by the balanced photo detectors is converted to an electrical signal:    After passing through the filter the field can be written: where erf stands for the Error function.
This optical signal after passing through the DI and then being detected by the balanced photo detectors is converted to an electrical signal, which is proportional to: 8 Therefore, at the maxima point (t=0) of the eye-pattern:  Mikkelsen et. al. [11] results, the squares stand for Malouin et. al. [12] simulations. The solid line results from the optimization of the eye-pattern (without noises); the dashed line stands for the analytical analysis , the dotted line represent formula (9) and the dash-dotted line represents the linear approximation (Eq.10).
A phenomenological approximation and linear one. The resultant curve allows us also to find a phenomenological approximation for the optimal FSR vs. filter's BW curve.
In Fig.7 The two functions (9) and (10) are qualitatively similar, but the difference between them diverges for narrow bandwidth (see Fig.7). This is not surprising since, as was explained at the beginning of the paper, the DI cannot be regarded as a linear filter.
Yet, since this effect is qualitatively insensitive of the filter shape, the linear approximation is a good approximation.
Summary. The effect, where larger FSR can improve BER for spectrally narrow channels, is investigated. It is shown that by optimizing the eye-opening of a noiseless signal an excellent estimation of the optimal FSR is achieved. We also find the exact curve of this effect by analyzing analytically the influence of the sub-sequences, which cause the effect. To the best of our knowledge this is the first time that this effect was addressed analytically. This analytical analysis yields an excellent match to the simulation. To complete the discussion, a simple formula was derived for the prediction of the best FSR for a given spectral BW channel.